bac-s-maths

Papers (167)
2025
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 5 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 7 bac-spe-maths__asie-sept_j1 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 6 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__metropole-sept_j1 4 bac-spe-maths__metropole-sept_j2 5 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 5 bac-spe-maths__polynesie-sept_j1 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4
2024
bac-spe-maths__amerique-nord_j1 5 bac-spe-maths__amerique-nord_j2 4 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 7 bac-spe-maths__asie_j2 4 bac-spe-maths__centres-etrangers_j1 5 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__metropole-sept_j1 4 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4 bac-spe-maths__suede 4
2023
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 5 bac-spe-maths__amerique-sud_j1 6 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 9 bac-spe-maths__centres-etrangers_j2 8 bac-spe-maths__europe_j1 4 bac-spe-maths__europe_j2 4 bac-spe-maths__metropole-sept_j1 7 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 8 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4 bac-spe-maths__reunion_j1 4 bac-spe-maths__reunion_j2 4
2022
bac-spe-maths__amerique-nord_j1 4 bac-spe-maths__amerique-nord_j2 4 bac-spe-maths__amerique-sud_j1 4 bac-spe-maths__amerique-sud_j2 4 bac-spe-maths__asie_j1 4 bac-spe-maths__asie_j2 4 bac-spe-maths__caledonie_j1 4 bac-spe-maths__caledonie_j2 4 bac-spe-maths__centres-etrangers_j1 4 bac-spe-maths__centres-etrangers_j2 4 bac-spe-maths__madagascar_j1 4 bac-spe-maths__madagascar_j2 4 bac-spe-maths__metropole-sept_j1 9 bac-spe-maths__metropole-sept_j2 4 bac-spe-maths__metropole_j1 4 bac-spe-maths__metropole_j2 4 bac-spe-maths__polynesie-sept 4 bac-spe-maths__polynesie_j1 4 bac-spe-maths__polynesie_j2 4
2021
bac-spe-maths__amerique-nord 5 bac-spe-maths__asie_j1 5 bac-spe-maths__asie_j2 5 bac-spe-maths__centres-etrangers_j1 9 bac-spe-maths__centres-etrangers_j2 7 bac-spe-maths__metropole-juin_j1 5 bac-spe-maths__metropole-juin_j2 5 bac-spe-maths__metropole-sept_j1 8 bac-spe-maths__metropole-sept_j2 5 bac-spe-maths__metropole_j1 5 bac-spe-maths__metropole_j2 5 bac-spe-maths__polynesie 5
2020
antilles-guyane 9 caledonie 5 metropole 9 polynesie 9
2019
amerique-nord 5 amerique-sud 6 antilles-guyane 5 asie 6 caledonie 3 centres-etrangers 6 integrale-annuelle 4 liban 9 metropole 5 metropole-sept 5 polynesie 5
2018
amerique-nord 5 amerique-sud 5 antilles-guyane 6 asie 4 caledonie 5 centres-etrangers 17 liban 6 metropole 3 metropole-sept 5 polynesie 7 pondichery 7
2017
amerique-nord 6 amerique-sud 5 antilles-guyane 6 asie 8 caledonie 6 centres-etrangers 8 liban 5 metropole 5 metropole-sept 4 polynesie 7
2016
amerique-nord 5 amerique-sud 6 antilles-guyane 10 asie 5 caledonie 6 centres-etrangers 8 liban 6 metropole 7 metropole-sept 4 polynesie 5 pondichery 6
2015
amerique-nord 4 amerique-sud 8 antilles-guyane 4 asie 7 caledonie 7 centres-etrangers 9 liban 5 metropole 7 metropole-sept 9 polynesie 6 pondichery 5
2014
amerique-nord 4 amerique-sud 7 antilles-guyane 5 asie 4 caledonie 7 centres-etrangers 7 liban 7 metropole 5 metropole-sept 5 polynesie 5 pondichery 4
2013
amerique-nord 5 amerique-sud 4 antilles-guyane 9 asie 5 caledonie 5 centres-etrangers 8 liban 4 metropole 5 metropole-sept 5 polynesie 4 pondichery 4
2007
integrale-annuelle2 6
2018 metropole

3 maths questions

Q1 Applied differentiation Existence and number of solutions via calculus View
In this exercise, the plane is equipped with an orthonormal coordinate system.
The curve with equation is represented below: $$y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 2 \right) .$$ This curve is called a ``catenary''.
We are interested here in ``catenary arcs'' delimited by two points of this curve that are symmetric with respect to the $y$-axis. Such an arc is represented on the graph below in solid line. We define the ``width'' and ``height'' of the catenary arc delimited by the points $M$ and $M^{\prime}$ as indicated on the graph.
The purpose of the exercise is to study the possible positions on the curve of the point $M$ with strictly positive abscissa so that the width of the catenary arc is equal to its height.
  1. Justify that the problem studied reduces to finding the strictly positive solutions of the equation $$( E ) : \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 4 x - 2 = 0$$
  2. Let $f$ be the function defined on the interval $[ 0 ; + \infty [$ by: $$f ( x ) = \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 4 x - 2 .$$ a. Verify that for all $x > 0 , f ( x ) = x \left( \frac { \mathrm { e } ^ { x } } { x } - 4 \right) + \mathrm { e } ^ { - x } - 2$. b. Determine $\lim _ { x \rightarrow + \infty } f ( x )$.
  3. a. Let $f ^ { \prime }$ denote the derivative function of $f$. Calculate $f ^ { \prime } ( x )$, where $x$ belongs to the interval $[ 0 ; + \infty [$. b. Show that the equation $f ^ { \prime } ( x ) = 0$ is equivalent to the equation: $\left( \mathrm { e } ^ { x } \right) ^ { 2 } - 4 \mathrm { e } ^ { x } - 1 = 0$. c. By setting $X = \mathrm { e } ^ { x }$, show that the equation $f ^ { \prime } ( x ) = 0$ has as its unique real solution the number $\ln ( 2 + \sqrt { 5 } )$.
  4. The sign table of the derivative function $f ^ { \prime }$ of $f$ is given below:
    $x$0$\ln ( 2 + \sqrt { 5 } )$$+ \infty$
    $f ^ { \prime } ( x )$-0+

    a. Draw up the variation table of the function $f$. b. Prove that the equation $f ( x ) = 0$ has a unique strictly positive solution which we denote by $\alpha$.
  5. Consider the following algorithm where the variables $a$, $b$ and $m$ are real numbers: \begin{verbatim} While $b - a > 0.1$ do: $m \leftarrow \frac { a + b } { 2 }$ If $\mathrm { e } ^ { m } + \mathrm { e } ^ { - m } - 4 m - 2 > 0$, then: $b \leftarrow m$ Else: $a \leftarrow m$ End If End While \end{verbatim} a. Before execution of this algorithm, the variables $a$ and $b$ contain respectively the values 2 and 3. What do they contain at the end of the algorithm execution? Justify the answer by reproducing and completing the table opposite with the different values taken by the variables at each step of the algorithm.
    $m$$a$$b$$b - a$
    231
    2.5
    $\ldots$$\ldots$$\ldots$

    b. How can we use the values obtained at the end of the algorithm in the previous question?
  6. The width of the Gateway Arch arc, expressed in metres, is equal to twice the strictly positive solution of the equation: $$\left( E ^ { \prime } \right) : \mathrm { e } ^ { \frac { t } { 39 } } + \mathrm { e } ^ { - \frac { t } { 39 } } - 4 \frac { t } { 39 } - 2 = 0$$ Give a bound for the height of the Gateway Arch.
Q2 4 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
Exercise 2 (4 points)
The flu virus affects each year, during the winter period, part of the population of a city. Vaccination against the flu is possible; it must be renewed each year.
Part A
A study conducted in the city's population at the end of the winter period found that:
  • $40 \%$ of the population is vaccinated;
  • $8 \%$ of vaccinated people contracted the flu;
  • $20 \%$ of the population contracted the flu.

A person is chosen at random from the city's population and we consider the events: $V$: ``the person is vaccinated against the flu''; $G$: ``the person contracted the flu''.
  1. a. Give the probability of event $G$. b. Reproduce the probability tree below and complete the blanks indicated on four of its branches.
  2. Determine the probability that the chosen person contracted the flu and is vaccinated.
  3. The chosen person is not vaccinated. Show that the probability that they contracted the flu is equal to 0.28.

Part B
In this part, the probabilities requested will be given to $10 ^ { - 3 }$ near.
A pharmaceutical laboratory conducts a study on vaccination against the flu in this city. After the winter period, $n$ inhabitants of the city are randomly interviewed, assuming that this choice amounts to $n$ successive independent draws with replacement. We assume that the probability that a person chosen at random in the city is vaccinated against the flu is equal to 0.4. Let $X$ be the random variable equal to the number of vaccinated people among the $n$ interviewed.
  1. What is the probability distribution followed by the random variable $X$?
  2. In this question, we assume that $n = 40$. a. Determine the probability that exactly 15 of the 40 people interviewed are vaccinated. b. Determine the probability that at least half of the people interviewed are vaccinated.
  3. A sample of 3750 inhabitants of the city is interviewed, that is, we assume here that $n = 3750$. Let $Z$ be the random variable defined by: $Z = \frac { X - 1500 } { 30 }$. We admit that the probability distribution of the random variable $Z$ can be approximated by the standard normal distribution. Using this approximation, determine the probability that there are between 1450 and 1550 vaccinated individuals in the sample interviewed.
Q3 5 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View
Exercise 3 (5 points)
The purpose of this exercise is to examine, in different cases, whether the altitudes of a tetrahedron are concurrent, that is, to study the existence of an intersection point of its four altitudes. We recall that in a tetrahedron MNPQ, the altitude from M is the line passing through M perpendicular to the plane (NPQ).
Part A: Study of particular cases
We consider a cube ABCDEFGH. We admit that the lines (AG), (BH), (CE) and (DF), called ``main diagonals'' of the cube, are concurrent.
  1. We consider the tetrahedron ABCE. a. Specify the altitude from E and the altitude from C in this tetrahedron. b. Are the four altitudes of the tetrahedron ABCE concurrent?
  2. We consider the tetrahedron ACHF and work in the coordinate system $(\mathrm{A} ; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. a. Verify that a Cartesian equation of the plane (ACH) is: $x - y + z = 0$. b. Deduce that (FD) is the altitude from F of the tetrahedron ACHF. c. By analogy with the previous result, specify the altitudes of the tetrahedron ACHF from the vertices $\mathrm{A}$, $\mathrm{C}$ and H respectively. Are the four altitudes of the tetrahedron ACHF concurrent?

In the rest of this exercise, a tetrahedron whose four altitudes are concurrent will be called an orthocentric tetrahedron.
Part B: A property of orthocentric tetrahedra
In this part, we consider a tetrahedron MNPQ whose altitudes from vertices M and N intersect at a point K. The lines (MK) and (NK) are therefore perpendicular to the planes (NPQ) and (MPQ) respectively.
  1. a. Justify that the line (PQ) is perpendicular to the line (MK); we admit likewise that the lines (PQ) and (NK) are perpendicular. b. What can we deduce from the previous question regarding the line (PQ) and the plane (MNK)? Justify the answer.
  2. Show that the edges [MN] and [PQ] are perpendicular.

Thus, we obtain the following property: If a tetrahedron is orthocentric, then its opposite edges are perpendicular in pairs.
Part C: Application
In an orthonormal coordinate system, we consider the points: $$\mathrm { R } ( - 3 ; 5 ; 2 ) , \mathrm { S } ( 1 ; 4 ; - 2 ) , \mathrm { T } ( 4 ; - 1 ; 5 ) \quad \text { and } \mathrm { U } ( 4 ; 7 ; 3 ) .$$ Is the tetrahedron RSTU orthocentric? Justify.